Smoothing and Global Attractors for the Zakharov System on the Torus

TL;DR

This paper studies the Zakharov system on a one-dimensional torus, demonstrating smoothing effects, polynomial bounds, and the existence of global attractors for the forced and damped case.

Contribution

It establishes smoothing properties and global attractors for the Zakharov system with periodic boundary conditions, including the forced and damped versions, in the energy space.

Findings

Difference between nonlinear and linear evolution is smoother over time.

Polynomial bounds for Sobolev norms above energy level.

Existence and smoothness of global attractors for the damped system.

Abstract

In this paper we consider the Zakharov system with periodic boundary conditions in dimension one. In the first part of the paper, it is shown that for fixed initial data in a Sobolev space, the difference of the nonlinear and the linear evolution is in a smoother space for all times the solution exists. The smoothing index depends on a parameter distinguishing the resonant and nonresonant cases. As a corollary, we obtain polynomial-in-time bounds for the Sobolev norms with regularity above the energy level. In the second part of the paper, we consider the forced and damped Zakharov system and obtain analogous smoothing estimates. As a corollary we prove the existence and smoothness of global attractors in the energy space.